Let $E$ be a collection of subsets in a space $X$. I would like to show that for all elements $A \in \sigma(E)$, the $\sigma$-ring generated by $E$, the exists a countable collection of subsets $E_i\in E$ such that $A \subset \cup_{i\ge1}E_i$.
I am trying to proceed by assuming, on the contrary, the existence of some $B\in\sigma(E)$ which is not covered by a countable union of such $E_i$, and showing the $\sigma(E)\setminus \{B\}$ is still a sigma ring containing $E$, but this isn't working. Is there a better way to proceed?